Subgroups — #27367930

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285292	C₂	1	1	yes	maximal, characteristic
2	#20284182	C₂	1	2	yes	maximal, complemented
3	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274158	C₂	1	2	yes	maximal, complemented
5	#20274157	C₂	1	2	yes	maximal, complemented
6	#20274110	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274109	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274072	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273472	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273389	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1668	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133246	—	2	2	no	—
13	#16811171	C₂²	1	2	yes	—
14	C₄×Q₈×D₅₃	C₂²	1	1	yes	characteristic
15	#16811028	C₂²	1	1	yes	characteristic
16	#16810969	C₂²	1	2	yes	—
17	#16810856	—	2	1	no	—
18	#16810856	—	2	1	no	—
19	#16810844	—	2	2	no	—
20	#16810843	—	2	2	no	—
21	#16810834	C₂²	1	2	yes	—
22	#16810833	C₂²	1	2	yes	—
23	#16810820	C₂²	1	2	yes	—
24	Q₈×#88322	C₂²	1	1	yes	characteristic
25	C₈×Dic₁₀₆	—	2	1	no	—
26	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
27	#16810747	—	2	1	no	—
28	#16810747	C₂²	1	1	yes	characteristic
29	D₈×Dic₅₃	—	2	1	no	—
30	D₈×Dic₅₃	—	2	1	no	—
31	#16810645	C₂²	1	2	yes	—
32	C₄⋊C₈×D₅₃	—	2	1	no	—
33	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
34	#16810621	—	2	2	no	—
35	C₄×C₈×D₅₃	—	2	1	no	—
36	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
37	#16810376	C₂²	1	1	yes	characteristic
38	#16810375	C₂²	1	2	yes	—
39	#16810370	C₂²	1	2	yes	—
40	#16810368	C₂²	1	2	yes	—
41	#16810367	C₂²	1	2	yes	—
42	C₅₃×#493	C₂²	1	1	yes	characteristic
43	C₅₃×#460	C₂²	1	2	yes	—
44	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#335	C₂²	1	1	yes	characteristic
46	C₅₃×#331	C₂²	1	2	yes	—
47	C₂×#11719947	—	2	2	no	—
48	C₂×Q₈×D₅₃	D₄	1	1	yes	characteristic
49	#13133246	—	4	8	no	—
50	C₂×Dic₂₁₂	—	2	2	no	—
51	C₂×#11719925	—	2	2	no	—
52	C₂×#11719923	—	2	2	no	—
53	C₂×#11719912	—	2	2	no	—
54	C₂×C₈×D₅₃	—	2	1	no	—
55	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
56	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
57	#13133217	C₂³	1	2	yes	—
58	#13133204	—	2	2	no	—
59	#13133188	—	2	2	no	—
60	#13133188	—	2	2	no	—
61	Q₈×Dic₅₃	D₄	1	1	yes	characteristic
62	C₄×Dic₁₀₆	—	2	1	no	—
63	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
64	#13133182	—	2	1	no	—
65	#13133182	D₄	1	1	yes	characteristic
66	D₄×Dic₅₃	—	2	2	no	—
67	D₄×Dic₅₃	—	2	2	no	—
68	C₄⋊C₄×D₅₃	—	2	1	no	—
69	C₄⋊C₄×D₅₃	D₄	1	1	yes	characteristic
70	C₄²×D₅₃	—	2	1	no	—
71	C₄²×D₅₃	C₂³	1	1	yes	characteristic
72	#13133151	—	2	1	no	—
73	#13133151	—	2	1	no	—
74	C₄⋊#88322	—	2	1	no	—
75	C₄⋊#88322	C₂³	1	2	yes	—
76	C₄⋊#88322	C₂³	1	1	yes	characteristic
77	#13133146	—	2	1	no	—
78	#13133146	—	2	2	no	—
79	#13133146	—	2	1	no	—
80	#13133146	—	2	1	no	—
81	#13133142	—	2	2	no	—
82	#13133140	—	2	2	no	—
83	#13133138	—	2	2	no	—
84	#13133138	—	2	2	no	—
85	#13133138	—	2	2	no	—
86	C₄×#88322	—	2	1	no	—
87	C₄×#88322	C₂³	1	1	yes	characteristic
88	C₈×Dic₅₃	—	2	1	no	—
89	C₈×Dic₅₃	—	2	1	no	—
90	C₈×Dic₅₃	D₄	1	1	yes	characteristic
91	C₈×Dic₅₃	D₄	1	1	yes	characteristic
92	C₂×SD₁₆×C₅₃	—	2	2	no	—
93	C₂×D₈×C₅₃	—	2	1	no	—
94	C₂×D₈×C₅₃	—	2	1	no	—
95	#127×C₅₃	C₂³	1	2	yes	—
96	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
97	C₄⋊C₈×C₅₃	—	2	1	no	—
98	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
99	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
100	#102×C₅₃	—	2	2	no	—
101	C₄×C₈×C₅₃	—	2	1	no	—
102	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
103	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂×D₄×C₅₃	—	2	2	no	—
105	C₂×D₄×C₅₃	—	2	2	no	—
106	C₂×D₄×C₅₃	—	2	2	no	—
107	C₂×C₂²⋊D₅₃	—	4	2	no	—
108	C₂²×Dic₅₃	—	4	2	no	—
109	Q₈×D₅₃	—	2	2	no	—
110	Q₈×D₅₃	D₈	1	4	yes	complemented
111	#11719947	—	4	4	no	—
112	#11719947	—	4	4	no	—
113	C₂×C₄×D₅₃	—	2	1	no	—
114	C₂×C₄×D₅₃	—	2	1	no	—
115	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
118	C₂×Dic₁₀₆	—	2	1	no	—
119	C₂×Dic₁₀₆	—	2	2	no	—
120	C₂×Dic₁₀₆	C₂×D₄	1	1	yes	characteristic
121	SD₁₆×C₅₃	—	4	4	no	—
122	D₈×C₅₃	—	4	2	no	—
123	D₈×C₅₃	—	4	2	no	—
124	C₂×C₈×C₅₃	—	2	1	no	—
125	C₂×C₈×C₅₃	—	2	2	no	—
126	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₄⋊C₄×C₅₃	—	2	1	no	—
129	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₅₃	—	2	1	no	—
131	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
132	C₂²⋊Dic₅₃	—	4	2	no	—
133	C₂²⋊Dic₅₃	—	4	2	no	—
134	#11719925	—	4	4	no	—
135	#11719923	—	4	4	no	—
136	C₄⋊Dic₅₃	—	2	1	no	—
137	C₄⋊Dic₅₃	—	2	2	no	—
138	C₄⋊Dic₅₃	—	2	2	no	—
139	C₄⋊Dic₅₃	C₂×D₄	1	1	yes	characteristic
140	#11719919	—	2	1	no	—
141	#11719919	—	2	1	no	—
142	#11719919	—	2	1	no	—
143	C₄×Dic₅₃	—	2	1	no	—
144	C₄×Dic₅₃	—	2	1	no	—
145	C₄×Dic₅₃	—	2	1	no	—
146	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
147	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
148	C₄×Dic₅₃	#41	1	1	yes	characteristic
149	C₂×#88322	—	2	1	no	—
150	C₂×#88322	—	2	2	no	—
151	C₂×#88322	C₂×D₄	1	1	yes	characteristic
152	C₂×#88322	C₂×D₄	1	1	yes	characteristic
153	Dic₂₁₂	—	4	4	no	—
154	#11719912	—	4	4	no	—
155	C₈×D₅₃	—	2	2	no	—
156	C₈×D₅₃	—	4	1	no	—
157	C₈×D₅₃	—	4	2	no	—
158	C₂³×C₅₃	—	4	2	no	—
159	C₂²×D₅₃	#121	1	1	yes	characteristic
160	Q₈×C₅₃	—	2	1	no	—
161	Q₈×C₅₃	C₂×D₈	1	2	yes	—
162	D₄×C₅₃	—	4	2	no	—
163	D₄×C₅₃	—	4	2	no	—
164	D₄×C₅₃	—	4	2	no	—
165	D₄×C₅₃	—	4	2	no	—
166	D₄×C₅₃	—	8	2	no	—
167	C₂×C₄×C₅₃	—	2	1	no	—
168	C₂×C₄×C₅₃	—	2	1	no	—
169	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
172	C₂²⋊D₅₃	—	8	4	no	—
173	C₂×Dic₅₃	—	2	1	no	—
174	C₂×Dic₅₃	—	2	1	no	—
175	C₂×Dic₅₃	—	4	2	no	—
176	C₂×Dic₅₃	—	8	4	no	—
177	C₂×Dic₅₃	#121	1	1	yes	characteristic
178	C₂×Dic₅₃	#121	1	1	yes	characteristic
179	C₂×Dic₅₃	#121	1	1	yes	characteristic
180	C₄×D₅₃	—	2	4	no	—
181	C₄×D₅₃	—	2	1	no	—
182	C₄×D₅₃	—	2	1	no	—
183	C₄×D₅₃	—	4	1	no	—
184	C₄×D₅₃	—	4	2	no	—
185	C₄×D₅₃	C₂×D₈	1	2	yes	—
186	C₄×D₅₃	#135	1	2	yes	—
187	Dic₁₀₆	—	2	1	no	—
188	Dic₁₀₆	—	2	4	no	—
189	Dic₁₀₆	—	4	2	no	—
190	Dic₁₀₆	—	4	2	no	—
191	Dic₁₀₆	C₂×D₈	1	2	yes	—
192	C₈×C₅₃	—	2	1	no	—
193	C₈×C₅₃	—	4	2	no	—
194	C₈×C₅₃	—	4	1	no	—
195	C₈×C₅₃	#137	1	2	yes	complemented
196	#88322	—	2	1	no	—
197	#88322	—	2	1	no	—
198	#88322	—	4	2	no	—
199	#88322	—	4	1	no	—
200	#19261	—	53	1	no	maximal, Sylow
201	C₂²×C₅₃	—	8	2	no	—
202	C₂²×C₅₃	—	8	2	no	—
203	C₂²×C₅₃	—	8	2	no	—
204	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
205	C₂×D₅₃	—	2	1	no	—
206	C₂×D₅₃	—	2	1	no	—
207	C₂×D₅₃	#466	1	2	yes	—
208	C₄×C₅₃	—	2	2	no	—
209	C₄×C₅₃	—	2	1	no	—
210	C₄×C₅₃	—	4	1	no	—
211	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
212	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
213	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
214	C₄×C₅₃	#575	1	1	yes	characteristic
215	Dic₅₃	—	2	1	no	—
216	Dic₅₃	—	2	2	no	—
217	Dic₅₃	—	2	1	no	—
218	Dic₅₃	—	4	1	no	—
219	Dic₅₃	—	8	2	no	—
220	Dic₅₃	#466	1	2	yes	—
221	#3169	—	53	1	no	—
222	#3037	—	53	2	no	—
223	C₂×C₈×Q₈	—	53	1	no	—
224	#1682	—	53	2	no	—
225	#1682	—	53	2	no	—
226	#1674	—	53	1	no	—
227	#1674	—	53	1	no	—
228	#1668	—	53	1	no	—
229	#1668	D₅₃	1	1	yes	characteristic, complemented
230	#1567	—	53	1	no	—
231	#1547	—	53	2	no	—
232	C₂×C₅₃	—	16	2	no	—
233	C₂×C₅₃	#3037	1	1	yes	characteristic
234	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
235	C₂×C₅₃	#3067	1	1	yes	characteristic
236	D₅₃	—	2	2	no	—
237	C₂×#137	—	106	2	no	—
238	#532	—	53	2	no	—
239	C₂×C₄×Q₈	—	53	1	no	—
240	#500	—	53	1	no	—
241	#493	C₂×D₅₃	1	1	yes	characteristic
242	#486	—	53	2	no	—
243	#466	—	106	1	no	—
244	#466	—	106	1	no	—
245	#464	—	106	2	no	—
246	#464	—	106	2	no	—
247	#462	—	53	2	no	—
248	#462	—	53	2	no	—
249	#460	—	53	2	no	—
250	#460	C₂×D₅₃	1	2	yes	—
251	C₈×Q₈	—	53	1	no	—
252	C₈×Q₈	—	53	1	no	—
253	C₈×Q₈	—	53	1	no	—
254	C₈×Q₈	—	106	1	no	—
255	C₈×Q₈	—	106	1	no	—
256	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
257	C₄×D₈	—	106	1	no	—
258	C₄×D₈	—	106	1	no	—
259	#423	—	53	2	no	—
260	C₂×C₄⋊C₈	—	53	1	no	—
261	C₂×C₄⋊C₈	—	106	1	no	—
262	#417	—	106	2	no	—
263	C₂×C₄×C₈	—	53	1	no	—
264	C₂×C₄×C₈	—	106	1	no	—
265	#335	—	53	2	no	—
266	#335	—	53	1	no	—
267	#335	C₂×D₅₃	1	1	yes	characteristic
268	#332	—	53	2	no	—
269	#332	—	53	2	no	—
270	#331	—	53	2	no	—
271	#331	C₂×D₅₃	1	2	yes	—
272	C₅₃	#19261	1	1	yes	characteristic, complemented, Sylow
273	C₂×#41	—	106	2	no	—
274	C₂²×Q₈	—	53	1	no	—
275	#137	—	212	8	no	—
276	C₂×Q₁₆	—	106	2	no	—
277	C₂×Q₁₆	—	106	2	no	—
278	C₂×SD₁₆	—	2	2	no	—
279	C₂×SD₁₆	—	106	2	no	—
280	C₂×D₈	—	2	1	no	—
281	C₂×D₈	—	2	1	no	—
282	C₂×M₁₆	—	106	2	no	—
283	C₂²×C₈	—	53	1	no	—
284	C₂²×C₈	—	53	1	no	—
285	C₂²×C₈	—	106	1	no	—
286	C₄⋊Q₈	—	53	2	no	—
287	#127	C₂²×D₅₃	1	2	yes	—
288	#124	—	106	2	no	—
289	#121	—	106	2	no	—
290	#121	—	106	2	no	—
291	C₄×Q₈	—	53	1	no	—
292	C₄×Q₈	—	53	1	no	—
293	C₄×Q₈	—	53	1	no	—
294	C₄×Q₈	—	106	1	no	—
295	C₄×Q₈	—	106	1	no	—
296	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
297	C₄×D₄	—	106	2	no	—
298	C₄×D₄	—	106	2	no	—
299	C₂×C₄⋊C₄	—	53	1	no	—
300	C₂×C₄⋊C₄	—	106	1	no	—
301	C₂×C₄²	—	53	1	no	—
302	C₂×C₄²	—	106	1	no	—
303	#107	—	106	1	no	—
304	#107	—	106	1	no	—
305	C₄⋊C₈	—	2	1	no	—
306	C₄⋊C₈	—	53	1	no	—
307	C₄⋊C₈	—	53	2	no	—
308	C₄⋊C₈	—	106	1	no	—
309	C₄⋊C₈	—	106	1	no	—
310	C₄⋊C₈	—	106	2	no	—
311	C₄⋊C₈	—	106	1	no	—
312	C₄⋊C₈	—	106	1	no	—
313	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
314	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
315	#103	—	106	2	no	—
316	#103	—	106	2	no	—
317	#102	—	2	2	no	—
318	#102	—	106	2	no	—
319	#102	—	106	2	no	—
320	#102	—	106	2	no	—
321	C₄×C₈	—	2	1	no	—
322	C₄×C₈	—	53	1	no	—
323	C₄×C₈	—	53	1	no	—
324	C₄×C₈	—	53	1	no	—
325	C₄×C₈	—	106	1	no	—
326	C₄×C₈	—	106	1	no	—
327	C₄×C₈	—	106	1	no	—
328	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
329	#41	—	212	4	no	—
330	#41	—	212	4	no	—
331	C₂×Q₈	—	53	4	no	—
332	C₂×Q₈	—	53	1	no	—
333	C₂×Q₈	—	106	1	no	—
334	C₂×Q₈	—	106	2	no	—
335	C₂×Q₈	—	106	2	no	—
336	C₂×Q₈	D₄×D₅₃	1	1	yes	characteristic
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	212	2	no	—
341	C₂²×C₄	—	53	1	no	—
342	C₂²×C₄	—	53	1	no	—
343	C₂²×C₄	—	53	1	no	—
344	C₂²×C₄	—	106	1	no	—
345	C₂²×C₄	—	106	1	no	—
346	C₂²×C₄	—	212	2	no	—
347	Q₁₆	—	212	4	no	—
348	Q₁₆	—	212	4	no	—
349	SD₁₆	—	4	4	no	—
350	SD₁₆	—	212	4	no	—
351	D₈	—	4	2	no	—
352	D₈	—	4	2	no	—
353	M₁₆	—	212	4	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	53	1	no	—
357	C₂×C₈	—	53	1	no	—
358	C₂×C₈	—	106	2	no	—
359	C₂×C₈	—	106	1	no	—
360	C₂×C₈	—	106	2	no	—
361	C₂×C₈	—	212	1	no	—
362	C₂×C₈	—	212	2	no	—
363	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
364	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	53	1	no	—
367	C₄⋊C₄	—	106	1	no	—
368	C₄⋊C₄	—	106	1	no	—
369	C₄⋊C₄	—	106	1	no	—
370	C₄⋊C₄	—	106	1	no	—
371	C₄⋊C₄	—	106	2	no	—
372	C₄⋊C₄	—	106	2	no	—
373	C₄⋊C₄	D₄×D₅₃	1	1	yes	characteristic
374	C₂²⋊C₄	—	212	2	no	—
375	C₂²⋊C₄	—	212	2	no	—
376	C₄²	—	2	1	no	—
377	C₄²	—	53	1	no	—
378	C₄²	—	53	1	no	—
379	C₄²	—	53	1	no	—
380	C₄²	—	106	1	no	—
381	C₄²	—	106	1	no	—
382	C₄²	—	106	1	no	—
383	C₄²	C₂³×D₅₃	1	1	yes	characteristic
384	C₂³	—	4	2	no	—
385	C₂³	—	53	1	no	—
386	Q₈	—	2	1	no	—
387	Q₈	—	53	2	no	—
388	Q₈	—	106	4	no	—
389	Q₈	—	106	1	no	—
390	Q₈	—	212	2	no	—
391	Q₈	—	212	2	no	—
392	Q₈	D₈×D₅₃	1	2	yes	—
393	D₄	—	4	2	no	—
394	D₄	—	4	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	2	no	—
397	D₄	—	8	2	no	—
398	D₄	—	424	4	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	53	1	no	—
402	C₂×C₄	—	53	2	no	—
403	C₂×C₄	—	53	2	no	—
404	C₂×C₄	—	53	1	no	—
405	C₂×C₄	—	53	1	no	—
406	C₂×C₄	—	106	4	no	—
407	C₂×C₄	—	106	1	no	—
408	C₂×C₄	—	106	1	no	—
409	C₂×C₄	—	106	1	no	—
410	C₂×C₄	—	106	1	no	—
411	C₂×C₄	—	424	4	no	—
412	C₂×C₄	—	212	1	no	—
413	C₂×C₄	—	212	2	no	—
414	C₂×C₄	—	212	2	no	—
415	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
416	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
417	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
418	C₈	—	2	1	no	—
419	C₈	—	4	2	no	—
420	C₈	—	4	1	no	—
421	C₈	—	106	1	no	—
422	C₈	—	106	1	no	—
423	C₈	—	212	2	no	—
424	C₈	—	212	1	no	—
425	C₈	#13133246	1	2	yes	complemented
426	C₂²	—	8	2	no	—
427	C₂²	—	8	2	no	—
428	C₂²	—	8	2	no	—
429	C₂²	—	53	2	no	—
430	C₂²	—	106	1	no	—
431	C₂²	—	106	1	no	—
432	C₂²	#121×D₅₃	1	1	yes	characteristic
433	C₄	—	2	1	no	—
434	C₄	—	2	2	no	—
435	C₄	—	4	1	no	—
436	C₄	—	53	2	no	—
437	C₄	—	106	1	no	—
438	C₄	—	106	1	no	—
439	C₄	—	106	2	no	—
440	C₄	—	424	2	no	—
441	C₄	—	212	1	no	—
442	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
443	C₄	#135×D₅₃	1	1	yes	characteristic
444	C₄	C₂×#13133246	1	1	yes	characteristic
445	C₄	#16811419	1	1	yes	characteristic
446	C₂	—	16	2	no	—
447	C₂	—	106	2	no	—
448	C₂	#20284182	1	1	yes	characteristic
449	C₂	#466×D₅₃	1	1	yes	characteristic
450	C₂	#20284433	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1668	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#20285292	C₂
hypercenter	#1668	D₅₃
hypocenter	C₅₃	#19261
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1668

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #20284182	C₂	semidirect
3. C₈×Q₈×D₅₃	C₂	semidirect
4. #20274158	C₂	semidirect
5. #20274157	C₂	semidirect
6. #20274110	C₂	semidirect
7. #20274109	C₂	semidirect
8. #20274072	C₂	semidirect
9. #20273472	C₂	semidirect
10. #20273389	C₂	semidirect
11. C₅₃×#1668	C₂	semidirect
110. Q₈×D₅₃	D₈	semidirect
195. C₈×C₅₃	#137	semidirect
229. #1668	D₅₃	semidirect
272. C₅₃	#19261	semidirect
425. C₈	#13133246	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20285292	2
#20284182	2
C₈×Q₈×D₅₃	2
#20274158	2
#20274157	2
#20274110	2
#20274109	2
#20274072	2
#20273472	2
#20273389	2
C₅₃×#1668	2
#19261	53